# A measure of strain at the micro and nano level

#### Three Measures of Strain

Engineering strain can be regarded in three ways:

**Type 1** is the macroscopic strain across the whole component – this is the type of strain imposed by external constraints whether as a consequence of it being a component of a machine, construction or device or as a result of an imposed strain during mechanical testing.

**Type 2** strain is internally generated but ranges over several grains such as produced by a fibre imbedded in a metal matrix. The effects will vary from grain to grain because of the elastic anisotropy of the metal and the varying orientation of the grains.

**Type 3** strain is the microscopic strain varying from point to point within each grain. Its components are any residual strain from mechanical loading, from type 2 strains and from strain due to incompatible lattice matching at grain boundaries, to precipitates within the grains and from the remaining dislocation structure. It is where the interaction of microstructure and the strain field occurs. This aspect of strain is increasingly useful for the analysis of fatigue and crack formation as well as in the analysis of functional materials (semiconductors, photo-voltaic cells, piezo-electric devices etc.) where the strain field can have a marked effect on the desired properties of the material.

CrossCourt uniquely allows the study and mapping of **type 3 elastic strains** (as well as the corresponding stress field) using data collected using a standard EBSD system attached to a regular Scanning Electron Microscope. It provides the strain distribution at each point within a grain relative to a selected point in the grain, usually that with the smallest KAM (Kernel Average Misorientation) value and hence the most likely point of minimum or zero strain.

#### Strain Measurement and HR EBSD

In High Resolution EBSD, high quality EBSD patterns from very similar regions of a crystalline material (e.g. from within the same grain of a metal) are compared and any differences between them due to lattice distortion, are measured using cross correlation based methods. These differences are then used to calculate the distortion matrix required to match the measured changes.

This calculation is based on work presented in:-

*A J. Wilkinson, G. Meaden & D. J. Dingley “High-resolution elastic strain measurement from electron backscatter diffraction patterns: New levels of sensitivity” Ultramicroscopy 106 2006, Pages 307–313.*

*doi:10.1016/j.ultramic.2005.10.001*

We use infinitesimal strain theory *(Nye book link*) to separate the distortion matrix into the elastic strain matrix and a set of rotations about the reference axes which describe how the lattice has rotated (*see the section on effects of plastic strain*).

The elastic strain tensor is a 3×3 matrix whose elements are the normal strains (tensile or compressive) parallel to three reference directions in the crystal and the shear strains acting along the three planes normal to the reference directions.

#### Refinements

##### Remapping

In samples where there is a significant amount of lattice rotation (i.e. greater than 1 degree), infinitesimal strain theory starts to break down and the elastic strain components cannot be extracted so easily (the figures are effectively polluted by the larger rotations).

To get ‘cleaner’ strain measurement, Crosscourt 4 can perform a second analysis of the data – called the remapping pass. Essentially the remapping pass uses the infinitesimal rotations measured in the first analysis to back rotate the recorded EBSD pattern (around the pattern centre) in such a way as to remove the measured rotations. These remapped patterns are then analysed (mostly) as before and the elastic strains can be extracted without the influence of the rotations.

##### Robust Fitting

Usually, more measurements of the pattern distortion (around 20 to 40) are made than are necessary to calculate the distortion matrix (4 are needed). As the problem is over determined a least squares solution is generally found.

However, in the cases of:-

- gross blurring of some Kikuchi bands in the pattern due to the presence of large number of statistically stored dislocations,
- marks or other phosphor defects,
- over saturation of the pattern in parts of the image,

some measures of the pattern distortion within a pattern may be better than others. In these cases, a robust fitting approach (iteratively weighted least squares) to the calculation of the distortion matrix (both before and after remapping) can vastly improve the measurement of elastic strain.

#### Effects of Plastic Strain

Plastic strain occurs when the stress is sufficiently high to operate dislocation sources, either from grain boundaries or from Frank Read sources leading to a rapid increase in the total number of dislocations distributed within the grains. The yield point of the material is defined as that where the multiplication of dislocations becomes very rapid. However, dislocation glide can occur prior to this macroscopic yield point so that the study of this early dislocation activity as well as what occurs a macroscopic yielding has been both of academic and engineering interest over a very large number of years.

HR EBSD can measure such dislocation activity because the strain field associated with a dislocation produces a lattice rotation which as mentioned above is part of and can be extracted from the deformation tensor. HREBSD is not sufficiently sensitive to measure the rotations of individual dislocations but it is sufficiently sensitive to detect the resulting rotation of a cluster of dislocations with a local density as small as ~10^{12 }m^{-2}. (Dislocations within this cluster whose Burgers vectors are of opposite sign have a resultant zero rotation and hence do not contribute to the observed rotation. Thus the measurements made by HR EBSD relate to only those dislocations within the clusters whose rotations do not mutually cancel. These are termed Geometrically Necessary Dislocations, GNDs). The highest densities that can be measured are in the ~10^{16 }m^{-2} range corresponding to dislocations 10nm apart. However, the spatial resolution of the HR EBSD is the same as that for conventional EBSD which is of the order 50nm so that in practice dislocation density distribution cannot be measured at a finer resolution than this. Thus the method yields the average value for GNDs within the area covered by the x and y scan step sizes. Finally, if the slip systems are known the analysis also provides a measure of the GND distribution of each slip component.

*J. Wilkinson & D. Randman 2010, “Determination of elastic strain fields and geometrically necessary dislocation distributions near nanoindents using electron back scatter diffraction” Phil. Mag. 90, (2010) Pages 1159-1177*

**http://dx.doi.org/10.1080/14786430903304145 **

#### Stress measurement

Distortion of the EBSD pattern which is a result of residual strain is what is measured experimentally. It is possible to calculate the stress state from these measurements through the standard relationship stress equals strain times Young modulus. In this case because both stress and strain are tensors, we need to use the full set of elastic stiffness coefficients in the calculation. A set of stresses equivalent to the strain set is calculated, i.e. three stress components acting parallel to the three references axes of the crystal and three shear stresses acting in the directions of the planes normal to the reference axes.

#### Engineering Applications

In engineering it is often more common to express the strains with reference to principal axes and the combined stresses in the form of the Von Mises stress yield criterion. The principal strains are those parallel to the three directions that remain unrotated under the influence of the residual strain one of which will be parallel to the maximum residual normal stress. The Von Mises yield criterion is a scalar in which the normal and shear stresses are combined in a manner that when the criterion reaches the yield stress of the material in simple tension then it is likely that the sample under the measured stress configuration will yield.